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On Klein & Milgram, concluded
Posted:
Aug 8, 2000 1:04 PM


I have just a few more comments about the paper on long division by David Klein and Jim Milgram.
1. costbenefit analysis.
K & M focus on the benefits of teaching the standard long division algorithm. I believe they exaggerate the necessity of standard long division as the sole means to achieve several of the benefits they cite. I also believe that many of the benefits they outline are rarely achieved in practice. They are theoretical, not real, goods. (Interestingly, the practical justification for teaching long division has been rendered largely obsolete by calculators. Anyone who has a practical need to do lots of long division problems is very unlikely to use any paperandpencil algorithm. K & M had to look elsewhere for reasons to retain the algorithm.)
But even if K & M are entirely correct in their claims about the benefits of teaching standard long division, their analysis is incomplete because they have failed to examine the costs of teaching this algorithm. (There is some arm waving about methods for teaching long division to children, but, as far as I know, these ideas are speculative and have not been tested in real classrooms. People have been trying to figure out a good way to teach standard long division for a long time, and my guess is that if there were one it would have been found out by now.)
It is impossible to judge whether it is worthwhile to teach the standard long division algorithm unless the costs are examined. More generally, other approaches to long division, with their costs and benefits, should also be examined. And this examination should not be purely a priori. Data, including student achievement data, should be considered.
I claim K & M seriously understimate the costs of teaching the standard long division algorithm. I am not talking about the costs when the algorithm is taught by an expert teacher under ideal conditions. I am talking about the costs when ordinary teachers teach the algorithm under typical conditions. Bringing all children to something like mastery takes months of time over several years. The approach typically taken to the standard long division algorithm is highly procedural and syntactic  in sharp contrast to the conceptual approach K & M clearly favor  which engenders misunderstanding of what mathematics is and consequent distaste for the subject.
Perhaps K & M meant their piece as a correction to reformers like Leinwand who have spent a lot of time identifying the high costs of teaching standard long division and other paperandpencil algorithms. If so, I believe they have overcorrected. A balanced assessment of costs and benefits would have been more helpful.
2. mathematics and mathematics education.
It would be helpful if K & M could at least pretend not to think that all math educators are idiots. It may be difficult for them to do this, but perhaps an atmosphere of mutual respect could be established that facilitate progress in our field.
Roughly 100 years ago, John Dewey pointed out that it's not enough to consider the subject matter that is being taught. One must also take the student into account. No doubt K & M have vastly greater knowledge of mathematics than most people in mathematics education. Certainly they know more mathematics than I do. But there are people in mathematics education who not only know some mathematics, but have actually taught school. And done research and read others' research. And written elementary school math textbooks. And know the history of mathematics education. The views of such people are not irrelevant to the questions whether and how long division should be taught. If K & M are uninterested in the discipline of mathematics education, I don't blame them. All I ask is that they recognize the limits of their expertise.
3. singleissue politics.
Probably what disturbs me most is the focus on long division as the sine qua non for a curriculum to be acceptable. Not only must paperandpencil long division be taught, but one certain algorithm must be taught. Any alternative is anathema.
I think our field is too complex to be reduced to such a litmus test.
Andy



